Intrinsic line density

The intrinsic density (also intrinsic carrier density ) is the characterizing material property of a semiconductor, in view of its electrical conductivity . It describes the charge carrier density of an ideal semiconductor that is undisturbed in its crystal lattice . The associated management mechanism is called intrinsic management .

The intrinsic conduction density determines the minimum value of the electrical conductivity. The actual conductivity of the semiconductor can be higher due to contamination or set to a higher value by doping . The conduction of impurities caused by this generally exceeds the conduction by several orders of magnitude at room temperature ; it thus covers the property of the base material.

Physical description

In semiconductors, all electrons are bound to the crystal atoms at absolute zero . Only when the temperature rises is sufficient thermal energy available to raise individual electrons to the energy level of the conduction band . The released electrons and the remaining defect electrons  ( holes ) are available for charge transport (generation of electron-hole pairs ); This effect is countered by the recombination of electron-hole pairs with the release of energy .

In thermodynamic equilibrium , there is an equilibrium between generation and recombination: the number density of freely moving charge carriers is constant over time. The intrinsic conduction density is thus composed of the average number density of free electrons and holes at the respective temperature: ${\displaystyle n_{\mathrm {i} }}$ ${\displaystyle n}$ ${\displaystyle p}$

Intrinsic conduction density for some important semiconductors ${\displaystyle T=300\;\mathrm {K} }$

semiconductor	${\displaystyle n_{\mathrm {i} }}$in cm −3
Germanium (Ge)	2.33e13 2.3e13
Silicon (Si)	1.0e10 1.5e10
Gallium arsenide (GaAs)	2.1e6 1.3e6th

${\displaystyle n_{\mathrm {i} }^{2}=n\cdot p}$(Law of mass action )

Arrhenius graph for the intrinsic conduction density of silicon as a function of the temperature according to the given equation

For lightly doped material, the Boltzmann distribution can be used instead of the Fermi distribution , then the charge carrier densities result as:

${\displaystyle n=N_{\mathrm {L} }\cdot \exp \left(-{\frac {E_{\mathrm {L} }-\mu }{k_{\mathrm {B} }\,T}}\right)}$

${\displaystyle p=N_{\mathrm {V} }\cdot \exp \left(+{\frac {E_{\mathrm {V} }-\mu }{k_{\mathrm {B} }\,T}}\right)}$

With

• ${\displaystyle n}$ :  Electron density
• ${\displaystyle p}$ : Hole density
• ${\displaystyle N_{\mathrm {L} }}$, : effective density of states in conduction and valence bands ${\displaystyle N_{\mathrm {V} }}$
• ${\displaystyle E_{\mathrm {L} }}$: Energy of the lower edge of the conduction band
• ${\displaystyle E_{\mathrm {V} }}$: Energy of the upper edge of the valence band
• ${\displaystyle E_{\mathrm {L} }-E_{\mathrm {V} }=E_{\mathrm {g} }}$: Band gap
• ${\displaystyle \mu }$ :  chemical potential
• ${\displaystyle k_{\mathrm {B} }}$: Boltzmann constant
• ${\displaystyle T}$ : absolute temperature

This gives the intrinsic conduction density:

${\displaystyle {\begin{aligned}n_{\mathrm {i} }&={\sqrt {n\cdot p{\color {White}|}}}\\&={\sqrt {N_{\mathrm {L} }\,N_{\mathrm {V} }}}\cdot \exp \left(-{\frac {E_{\mathrm {L} }-E_{\mathrm {V} }}{2\,k_{\mathrm {B} }\,T}}\right)\\&={\sqrt {N_{\mathrm {L} }\,N_{\mathrm {V} }}}\cdot \exp \left(-{\frac {E_{\mathrm {g} }}{2\,k_{\mathrm {B} }\,T}}\right)\\\end{aligned}}}$

This result is significantly dependent on the temperature . The corresponding diagram does not take into account that , and also depend on the temperature, but not as pronounced. In the area of ​​room temperature, it roughly doubles when the temperature increases by 10 K. ${\displaystyle N_{\mathrm {L} }}$ ${\displaystyle N_{\mathrm {V} }}$ ${\displaystyle E_{\mathrm {g} }}$ ${\displaystyle n_{\mathrm {i} }}$

State in real crystals

It should be noted that there is no such thing as a perfect crystal ( entropy argument of thermodynamics ). At an atomic density of about 5e²²  cm ⁻³ (guide value for silicon and germanium) would result in an impurity density of 5e¹³  cm ^{−3 means} that there is a single disturbance in the crystal lattice for every billion atoms of the semiconductor. But even a crystal that is so poor in defects is not yet poor enough for the state of intrinsic conduction. Conversely,the charge carrier density and thus the conductivitycan be increased by extrinsic impurities (doping). The typical density of the introduced foreign atoms is 10 ¹⁴ −10 ¹⁸  cm ⁻³ .

Since the degree of ionization of the impurities increases with the temperature, the charge carrier density initially increases with the temperature ( impurity reserve ). At room temperature (in the case of silicon) all impurities are normally ionized ( impurity depletion ), and the charge carrier density no longer depends on the temperature, but on the doping concentration; this case is called extrinsic conductivity .

If the temperature is increased further, the semiconductor loses its character as n-doped or p-doped, since more and more charge carriers are generated by the intrinsic charge carrier generation. The semiconductor material becomes intrinsically conductive because the thermal energy is now sufficient to excite electrons from the valence band to the conduction band to a greater extent. The practical use of doped semiconductor components is therefore only sufficient for germanium up to 70 ° C and for silicon up to about 170 ° C or in a temperature range typically −40 ° C   <<160 ° C. A calculation for the intrinsic conduction density of silicon at 700 K comes to 6 ${\displaystyle \vartheta }$.3e¹⁵  cm⁻³.

Individual evidence

1. ↑ Collaboration: Authors and Editors of the LB Volumes III / 17A-22A-41A1b: Germanium (Ge), intrinsic carrier concentration. In: O. Madelung, U. Rössler, M. Schulz (Eds.): SpringerMaterials - The Landolt-Börnstein Database. doi : 10.1007 / 10832182_503 .
2. ↑ ^{a b c d} Leonhard Stiny: Active electronic components. 4th edition. Springer Vieweg, 2019, pp. 28–30.
3. ↑ Collaboration: Authors and Editors of the LB Volumes III / 17A-22A-41A1b: Silicon (Si), intrinsic carrier concentration. In: O. Madelung, U. Rössler, M. Schulz (Eds.): SpringerMaterials - The Landolt-Börnstein Database. doi : 10.1007 / 10832182_455 .
4. ^ Pietro P. Altermatt, Andreas Schenk, Frank Geelhaar, Gernot Heiser: Reassessment of the intrinsic carrier density in crystalline silicon in view of band-gap narrowing . In: Journal of Applied Physics . tape 93 , no. 3 , February 2003, ISSN  0021-8979 , p. 1598-1604 , doi : 10.1063 / 1.1529297 .
5. ↑ Collaboration: Authors and Editors of the LB Volumes III / 17A-22A-41A1b: Gallium arsenide (GaAs), intrinsic carrier concentration. In: O. Madelung, U. Rössler, M. Schulz (Eds.): SpringerMaterials - The Landolt-Börnstein Database. doi : 10.1007 / 10832182_196 .
6. ^ ^{A b} Jürgen Eichler: Physics: Basics for engineering studies. 2nd Edition. Vieweg, 2004, p. 273 f.
7. ↑ Wolfgang Böge, Wilfried Plaßmann (ed.): Vieweg Handbook Electrical Engineering: Fundamentals and applications for electrical engineers. 3. Edition. Vieweg, 2004, p. 320.
8. ^ H. Hartl, E. Krasser, W. Pribyl, P. Söser, G. Winkler: Electronic circuit technology. Pearson, 2008, p. 104.
9. ↑ Helmut Lindner: Ground plan of solid state physics. Vieweg, 1979, p. 111.
10. ↑ Leonhard Stiny: Active electronic components. 4th edition. Springer Vieweg, 2019, p. 37.
11. ↑ Joachim Specovius: Basic course in power electronics: components, circuits and systems. 3. Edition. Vieweg + Teubner, 2009, p. 7.